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Testing for unit roots. II. (English) Zbl 0579.62014
Summary: [For part I see ibid. 49, 753-779 (1981; Zbl 0468.62021).]
This paper investigates the exact sampling distribution of the least squares estimator of $$\beta$$ in the model $$y_ t=\mu +\beta y_{t- 1}+u_ t$$ where the $$u_ t$$ are independently $$N(0,\sigma^ 2)$$. The distribution is calculated for the case where $$y_ 0$$ is a known constant and where $$y_ 0$$ is a random variable.
Given $$y_ 0$$ is a constant we prove a small $$\sigma$$ asymptotic result and compute the exact powers of nonsimilar tests of the random-walk hypothesis $$\beta =1$$ and of the stability hypothesis $$\beta =0.9$$. The exact powers of a test of the stability hypothesis are calculated for the case where $$y_ 0$$ is random. The accuracy of the standard normal approximation is examined for both start-up regimes.

##### MSC:
 62E15 Exact distribution theory in statistics 62P20 Applications of statistics to economics 62F03 Parametric hypothesis testing
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